Evaluate ${\left( {102} \right)^3}$ using the appropriate identities.

Hint: In the given question, we have to evaluate the cube of a number given to us in the problem itself with the help of a suitable algebraic identity. The base number given to us is $102$ and the exponent is $3$. So, we can use the algebraic identity ${\left( {a + b} \right)^3} = {a^3} + 3ab\left( {a + b} \right) + {b^3}$. The algebraic identity ${\left( {a + b} \right)^3} = {a^3} + 3ab\left( {a + b} \right) + {b^3}$ is used to evaluate the cube of a binomial expression involving the sum of two terms. So, to calculate the cube of the given number using the identity, we first split the number into two parts such that the number remains unchanged and then apply the identity.

Complete step by step answer:
Given question requires us to find the value of a cube of $102$ using a suitable identity. So, we can use the algebraic identity ${\left( {a + b} \right)^3} = {a^3} + 3ab\left( {a + b} \right) + {b^3}$ for computing the whole cube of a binomial expression. So, to calculate the cube of $102$, we have to first divide the number $102$ into two parts such that the following calculation of the square of the number becomes easier. So, we know that $102 = 100 + 2$. So, we can split $102$ into the two numbers $100$ and $2$. So, we now substitute the values of the two parts into the algebraic identity that we are supposed to use. Hence, we have, ${102^3} = {\left( {100 + 2} \right)^3}$

Now, we expand the right side of the equation using the algebraic identity to evaluate the cube of a binomial expression involving the sum of two terms. So, we get,
$ \Rightarrow {102^3} = {\left( {100} \right)^3} + 3\left( {100} \right)\left( 2 \right)\left( {100 + 2} \right) + {\left( 2 \right)^3}$
Now, we evaluate the cubes of the terms and carry out the calculations. So, we get,
$ \Rightarrow {102^3} = 1000000 + 600\left( {102} \right) + 8$
$ \Rightarrow {102^3} = 1000000 + 61200 + 8$
Now, adding up the terms, we get,
$ \therefore {102^3} = 10,61,208$

So, the value of ${102^3}$ calculated using the algebraic identity ${\left( {a + b} \right)^3} = {a^3} + 3ab\left( {a + b} \right) + {b^3}$ is $1061208$.

Note: Before attempting such questions, one should memorize all the algebraic identities and should know their applications in such problems. Care should be taken while carrying out the calculations. We can also verify the answer of the given question by calculating the cube of $102$ manually or using a calculator.